It has to do with how much "r" has to increase before the next "period doubling" (e.g. 1 --> 2, 2 -->4). Micheal Feigenbaum discovered that the ratio of the period doubling length from one period doubling to the next converges on 4.669… That's the Feigenbaum constant.
Map the distances between period doubling incidents, and the picture looks like this. It's a picture of increasingly elaborate results that eventually exhibits chaotic behavior.
